J. Chem. SOC., Faraday Trans. 1, 1982, 78, 185-195 Thermal Conductivity of Binary Gaseous Mixtures Containing Diatomic Components BY MARC J. ASSAEL AND WILLIAM A. WAKEHAM* Department of Chemical Engineering and Chemical Technology, Imperial College, Prince Consort Road, London SW7 2BY Received 3rd February, 198 1 Absolute measurements of the thermal conductivity of binary mixtures of hydrogen +nitrogen, helium +carbon monoxide and argon +carbon monoxide are reported. The measurements were performed in a transient hot-wire instrument at a temperature of 35 OC and in the pressure range 1.8-8 MPa; the reported data have an estimated uncertainty of +0.2%. The experimental results for hydrogen +nitrogen have been interpreted with the aid of the semi-classical kinetic theory expressions of Monchick, Pereira and Mason.It has been found that the theoretical relationship is not as accurate as the experimental data although, if more accurate theory were available, rotational collision numbers for relaxation of nitrogen by hydrogen could be derived from the measurements. A recently proposed scheme for the evaluation of the effect of density on the thermal conductivity of gas mixtures provides a satisfactory description of the experimentally observed first density coefficient in the density expansion of the thermal conductivity. The modern version of the transient hot-wire technique provides a means of determining the thermal conductivity of gases with an uncertainty of &0.2%. In a number of recent publications the technique has been applied to the pure monatomic gases and some of their mi~tures,l-~ to pure polyatomic g a ~ e s ~ - ~ and to some of their binary mixtures with monatomic specie^.^? The measurements have been carried out over a range of densities and near to room temperature.In the limit of zero density the measurements upon systems containing polyatomic components are probably the more important because the kinetic theory underlying the description of the thermal conductivity is less well-developed than for the monatomic species and is essentially untested against accurate experimental data. However, at elevated densities theories for both monatomic and polyatomic species are less than complete and the new data provide one means of assessing their current status. In this paper we extend our studies to two further systems involving diatomic and monatomic components (mixtures of helium and argon with carbon monoxide) and to a system involving two diatomic molecules (mixtures of hydrogen with nitrogen).The unexpected differences which have been observed between the thermal conductivity of pure carbon monoxide and that of pure nitrogen mean that the former systems contribute to a study which is analogous to that of Fleeter et aL7 concerning mixtures of nitrogen with the monatomic gases. The latter system represents the simplest type of mixture involving two diatomic species because the very slow relaxation of the rotational energy of hydrogen implies that only two relaxation times in the mixture are significant. 185 FAR 1186 THERMAL CONDUCTIVITY OF BINARY GAS MIXTURES EXPERIMENTAL A description of the transient hot-wire instrument employed for the measurements as well as of the necessary working equations has been given e1sewhere.l' 9* lo The instrument was used without alteration for the measurements on the hydrogen + nitrogen mixtures but prior to the measurements on the mixtures containing carbon monoxide a new set of platinum wires was installed in the cells.The lengths of the new wires were 1, = 161.11 mm and 1, = 63.53 mm for the long and short wire, respectively. In order to confirm the continued correct operation of the instrument following this change, the thermal conductivities of helium and argon were remeasured. In no case did the results of these repeated measurements depart by more than +0.2% from the correlation of the original data.' The measurements reported here, on three mixtures of hydrogen with nitrogen and two mixtures each of helium and argon with carbon monoxide, were carried out at 35 OC and in the pressure range 1.8-8 MPa. The mixtures were manufactured gravimetrically from pure gases supplied by Matheson and British Oxygen Ltd; the stated purity of the gases was in no case worse than 99.995%.The density of the mixtures at the equilibrium temperature of the cells, T,, was determined directly in situ as described earlier and the small correction of this density to the reference temperat~re,~ T,, was applied with the aid of the available pVT data.ll The uncertainty in the reported density is estimated to be of the order of & 0.3 % in the worst case.TABLE 1 .-THERMAL CONDUCTIVITY OF HYDROGEN + NITROGEN MIXTURES AT T,,, = 35 OC, xH2 = 0.7338 2.23 2.82 3.41 4.00 4.65 5.38 6.25 7.27 30.49 30.48 30.48 30.46 30.47 30.50 30.44 30.46 8.03 10.09 12.35 14.26 16.40 19.06 22.01 25.36 35.18 35.09 35.02 34.94 34.89 34.87 34.75 34.63 7.91 9.95 12.17 14.05 16.17 18.80 21.70 25.02 116.1 116.3 116.4 116.7 116.9 117.2 117.4 117.7 1 16.0 116.3 116.4 116.7 116.9 117.2 117.5 117.8 (i3A/i3T)350c = 0.368 mW m-l K-2. RESULTS Tables 1-3 contain the experimental thermal conductivity data for mixtures of hydrogen and nitrogen with mole fractions xH2 = 0.7338, xH2 = 0.4865 and xHq = 0.2136, respectively. Tables 4 and 5 contain the results for mixtures of helium +carbon monoxide with xHe = 0.7403 and xHe = 0.2404, respectively, whereas tables 6 and 7 list the data for the argon+carbon monoxide mixtures with xAr = 0.7036 and xAr = 0.4080, respectively.In each case, in addition to the thermal conductivity at the reference temperature A( q, pr), we report the thermal conductivity corrected to a nominal temperature, Trio, = 35 OC. The correction to the nominal temperature has been accomplished with the aid of the equation For the purpose of this correction, it has been assumed that the derivative (8A/8T),M. J. ASSAEL AND W . A. WAKEHAM 187 TABLE 2.-THERMAL CONDUCTIVITY OF HYDROGEN + NITROGEN MIXTURES AT T,,, = 35 OC, xHZ = 0.4865 1.66 2.13 2.46 3.26 3.92 4.58 5.30 6.14 7.16 30.50 30.46 30.5 1 30.54 30.52 30.58 30.52 30.55 30.48 10.54 13.23 16.55 20.14 24.12 28.02 32.39 37.34 43.21 35.42 35.23 35.25 35.12 35.09 35.07 34.89 34.88 34.71 10.38 13.03 16.30 19.84 23.77 27.62 3 1.94 36.83 42.63 74.80 74.83 74.98 74.99 75.08 75.13 75.31 75.50 75.71 74.69 74.77 74.92 74.96 75.06 75.1 1 75.34 75.53 75.78 TABLE 3.-THERMAL CONDUCTIVITY OF HYDROGEN+NITROGEN MIXTURES AT TnOm = 35 OC, xHz = 0.2136 2.02 2.72 3.33 3.88 4.68 5.41 6.28 7.31 30.51 30.46 30.50 30.50 30.47 30.51 30.46 30.49 18.13 24.67 29.99 34.89 41.89 48.26 55.88 64.7 1 35.29 35.06 35.02 34.95 34.83 34.79 34.62 34.60 17.86 24.3 1 29.56 34.40 41.31 47.60 55.14 63.86 44.72 44.99 45.17 45.22 45.52 45.73 46.06 46.35 ~~~ ~ 44.68 44.98 45.17 45.23 45.54 45.77 46.12 46.4 1 (aA/aT)35oc = 0.150 mW m-' K-2.TABLE 4.-THERMAL CONDUCTIVITY OF HELIUM + CARBON MONOXIDE MIXTURES AT TnOm = 35 OC, x H e = 0.7403 2.20 2.86 3.48 4.17 5.00 5.69 6.50 7.41 30.34 30.33 30.33 30.33 30.30 30.28 30.24 30.19 9.12 11.52 13.96 16.72 19.97 22.51 25.43 28.85 34.82 34.69 34.64 34.56 34.47 34.42 34.3 1 34.3 1 8.99 11.36 13.76 16.49 19.71 22.21 25.10 28.47 93.51 93.82 93.89 94.04 94.38 94.74 94.86 94.24 93.56 93.90 93.99 94.16 94.52 94.90 95.05 95.42 (aA/aT),,., = 0.274 mW m-' K-2.7-2188 THERMAL CONDUCTIVITY OF BINARY GAS MIXTURES TABLE 5.-THERMAL CONDUCTIVITY OF HELIUM + CARBON MONOXIDE MIXTURES AT Tnom = 35 O C , xHe = 0.2404 2.21 2.86 3.58 4.42 5.04 5.73 6.56 7.48 30.73 30.75 30.73 30.75 30.67 30.51 30.38 30.37 19.69 25.13 31.77 38.85 44.35 50.28 57.08 65.02 34.96 34.88 34.76 34.70 34.53 34.31 34.09 34.04 19.42 25.08 31.37 38.37 43.67 49.67 56.41 64.26 40.09 40.30 40.56 40.80 40.97 41.17 41.46 41.81 40.10 40.32 40.59 40.84 41.03 41.26 41.58 41.94 (i3A/i3T)350c = 0.130 mW m-l KP2.TABLE 6.-THERMAL CONDUCTIVITY OF ARGON + CARBON MONOXIDE MIXTURES AT Tnom = 35 O C , x A r = 0.7036 2.14 2.96 3.67 4.35 5.08 5.76 6.54 7.55 30.27 31.00 30.28 43.64 30.25 54.24 30.23 64.77 30.23 74.45 30.23 85.87 30.22 97.3 1 30.22 110.6 35.30 35.08 34.90 34.74 34.61 34.56 34.40 34.35 30.50 42.97 53.44 63.84 73.41 84.68 96.01 109.1 2 1.09 21.43 21.69 21.97 22.27 22.54 22.89 23.27 2 1.08 21.43 21.69 21.98 22.29 22.56 22.92 23.30 (dA/aT)350, = 0.053 mW m-l K-2. TABLE 7.-THERMAL CONDUCTIVITY OF ARGON + CARBON MONOXIDE MIXTURES AT TnOm = 35 O C , x A r = 0.4080 1.84 2.64 3.25 3.98 4.51 5.11 5.79 6.40 7.37 30.25 30.30 30.30 30.29 30.28 30.27 30.27 30.27 30.25 24.35 35.12 42.98 52.80 59.93 68.13 77.27 87.40 98.97 36.18 35.90 35.74 35.57 35.48 35.29 35.18 35.07 34.94 23.89 34.49 42.23 51.91 58.94 67.03 76.05 86.06 97.49 23.28 23.50 23.78 24.13 24.37 24.64 24.89 25.14 25.59 23.21 23.45 23.74 24.09 24.34 24.62 24.88 25.14 25.59 (i3A/i3T)350c = 0.056 mW m-l K-2.M.J. ASSAEL A N D W. A. WAKEHAM 189 is independent of density and equal to the derivative in the limit of zero density. The derivative itself has been estimated by means of the Hirschfelder-Eucken equation12 and experimental viscosity data for the mixtures of hydrogen and nitrogen.13 In the cases of the mixtures involving carbon monoxide viscosity data are not available and the derivative has therefore been obtained from the Hirschfelder-Eucken equation with the aid of estimates for the parameters of a Lennard-Jones (12-6) potential characteristic of the unlike interaction.In any event, because the correction does not exceed 0.2%, the additional uncertainty introduced into the reported thermal conductivity is negligible. The entire set of experimental thermal conductivity data for each gas mixture at the nominal temperature has been correlated by means of an equation of the form A@) = a, -+ alp + a&. (2) TABLE 8.-cOEFFICIENTS IN THE CORRELATING POLYNOMIAL OF EQN (2) FOR THE DENSITY DEPENDENCE OF THE THERMAL CONDUCTIVITY OF THE GAS MIXTURES a,lmW allPW a2lnW std. dev., m2 kg-l K-l m5 kg-2 K-l 0 (%) mixture ,-1 K-1 H2+N2 XHn - - 0.2136 44.02 37.3 xHo = 0.4865 74.33 32.3 x H ~ - 0.7338 115.2 105.5 xHe = 0.2404 39.45 32.0 He + CO xHe = 0.7403 92.72 94 Ar + CO xAr = 0.4080 22.39 32.6 xAr = 0.7036 20.28 25.3 - k 0.09 - f 0.06 - f 0.02 101 f 0.05 - + 0.08 - - k0.15 21 f 0.08 The coefficients a,, a, and a, which secure the best representation of this kind are collected in table 8, together with the standard deviation of the fit.Fig. 1 and 2 contain plots of the deviations of the experimental results from the correlation for hydrogen + nitrogen and the systems containing carbon monoxide, respectively. In neither case do the deviations from the correlation exceed &0.25%. The experimental data have been subjected to the statistical analysis described elsewhere14 in order to determine best estimates for the zero-density thermal con- ductivity of the mixtures, A,, as well as of the first density coefficient, c,, in the (3) expansion The values thus derived are listed in table 9, together with their standard deviation and the corresponding values for the pure gases obtained earlier.6 In the cases where a quadratic representation of the data according to eqn (2) is necessary, the coefficients a, and c, of tables 8 and 9 do not quite overlap within their standard deviations.This is because only two or three points lie within the range requiring a quadratic fit so that the statistical basis for the error estimate is not reliable. A = A0+c1p+czp2+. . . .190 O 0.2 41 X tl 8 x n - --- 8 c I K 6 c i '2 -0.2 Y .M THERMAL CONDUCTIVITY OF BINARY GAS MIXTURES I 1 8 0 8 . 0 ' 8 A A 0 0.: A m 0 A A I I FIG. 1.-Deviations of the experimental thermal conductivity data from the correlation of eqn (2) for mixtures of hydrogen+nitrogen: 0, x H , = 0.2136; ., xH, = 0.4865; A, xH, = 0.7338.0.4 &? 0 X 2 0.2 3 t: x n 8 8 1 c 7 0 * 5 6 Y c .- U ?2 -0.2 -0 -0.4 0 O o 0 0 8 . 0 0 8 0 0 0 I I 25 50 density, plkg m-3 FIG. 2.-Deviations of the experimental thermal conductivity data from the correlation of eqn (2) for mixtures of helium+carbon monoxide: 0, xHe = 0.2404; ., xHe = 0.7403; and for mixtures of argon+carbon monoxide: 0, xAr = 0.4080; 0, xAr = 0.7036.M. J. ASSAEL A N D W. A. WAKEHAM 191 TABLE 9.-BEST ESTIMATES FOR THE ZERO-DENSITY THERMAL CONDUCTIVITY, &, AND THE FIRST DENSITY COEFFICIENT, c,, FOR THE PURE GASES AND MIXTURES AT T,,,, = 35 "c H2+N2 XHn - 0.0 xH2 = 0.2136 xHp - 0.7338 xHZ = 0.4865 XHZ = 1.0 He + CO XHe = 0.0 xHe = 0.2404 xHe = 0.7403 XHe 1.0 Ar + CO XAr = 0.0 xAr = 0.4080 xA,.= 0.7036 XAr = 1.0 26.45 f 0.06 44.02 & 0.04 74.33 f 0.04 1 1 5.21 k 0.03 192.2k0.1 25.68 _+ 0.03 39.34 f 0.03 92.72 & 0.08 158.4 k 0.1 25.68 +_ 0.03 22.39 +_ 0.04 20.20 k 0.02 18.18 f 0.01 37f3 37k 1 32&2 106k2 950 f 30 42_+ 1 39.3 4 0.6 94+4 256 9 42+ 1 32.6 f 0.6 28.2 k 0.3 24.0 f 0.3 DISCUSSION Z E RO-D E N S I T Y THERMAL CONDUCT I V I T Y The most accurate available theoretical formulation for the thermal conductivity of a dilute, binary gas mixture containing polyatomic components is that given by Monchick et aLl2 The thermal conductivity may be written in the form where the first term, Amix(mon), represents the contribution to the thermal conductivity of the mixture from translational energy whereas the second term is the contribution from the transport of internal energy.Together, these two terms constitute the Hirschfelder-Eucken expression for the thermal conductivity.12 The final term, AA, incorporates the explicit effects of inelastic collisions into the theory and is given by a rather cumbersome expression, derived by Monchick et u1.,12 which is omitted here in view of its length. In eqn (4) the symbol At represents the thermal conductivity of pure gas i, Ai(mon) the translational contribution to it and xi the mole fraction of that component in the mixture. The quantity Diint, is the diffusion coefficient for internal energy of component i in component j . Eqn (4) is, essentially, a first-order, semi-classical kinetic theory approximation to the thermal conductivity.It therefore fails to account for any effects arising from spin polarization or from higher-order kinetic theory approximations. At present, the most accurate calculations with eqn (4) must make use of high-quality experimental viscosity data. Therefore, we confine the discussion here to the system hydrogen + nitrogen, which is the only one for which the necessary information is available. Furthermore, because we are interested in the ability of eqn (4) to describe the composition dependence of the thermal conductivity, we identify Ai with the192 THERMAL CONDUCTIVITY OF BINARY GAS MIXTURES experimental value of the pure gas thermal conductivity obtained earlier.’? In this way it is ensured that the data at the end points of the composition range are exactly reproduced by the calculation. The translational contributions to the thermal conductivity of pure hydrogen and nitrogen, as well as the interaction thermal conductivity llij(mon), which occurs in the evaluation of Amix(mon), have been derived from viscosity measurements on the pure gases and their mixtures by means of the relation12*15 15(m, + mj) I,(mon) = 8m,rnj k V i j where qij is the interaction viscosity for i # j , k is Boltzmann’s constant and m, is the mass of a molecule.The same viscosity data have also been used to obtain the diffusion coefficients for internal energy, assuming that they are identical to the mass diffusion There is some evidence that the equality of the diffusion coefficients for internal energy and mass is not exact.ls Nevertheless, the departures from equality are usually smalls and there is no theoretical guidance towards a better assignment so that eqn (6) represents the best possible scheme for estimation at present. The quantity A& which is found in eqn (6), is a ratio of two collision integralsl5 which are functionals of the intermolecular pair potential for the species i andj.Both AS and l?;, a similar collision integral ratio which is found in Lmix(mon), implicitly contain the effects of inelastic collisions in the gas of which account cannot properly be taken at presenf.l5Js These two quantities have therefore been estimated on the basis of elastic collisions only using a Lennard-Jones (1 2-6) potential for the nitrogen + hydrogen system suggested by Kestin and Yata,13 from the potential proposed by Gengenbach et all7 for hydrogen and from the extended law of corresponding states correlation for nitrogen.The remaining quantities required for the evaluation of the thermal conductivity of the mixture occur in the inelastic term All. They are the internal heat capacities of the components and the four rotational relaxation collision numbers, Cij, which quantify the number of collisions necessary for the equilibration of the internal (rotational) energy of the molecules of species i by collision with speciesj.12 The heat capgcity of nitrogen has been taken from the tables of Hilsenrath et al.19 and that of hydrogen from the work of McCarty.20 The rotational collision numbers of nitrogen and hydrogen have been determined by extrapolation of the low-temperature experimental data obtained by Prangsma et aL2’ and Jonkman et a1,22 respectively.In the case of the unlike interactions there are no direct measurements of the collision numbers so that following Monchick et a1.12 it has been necessary to equate them with those for the corresponding pure components. The numerical values of the quantities finally employed in the calculations are collected in table 10. It should be remarked that, in the case of hydrogen, the very slow relaxation of internal energy provides a sound basis for the use of elastic values of A* and B* as well as for the identity of the mass diffusion coefficient with the diffusion coefficient for internal energy according to eqn (6).Furthermore, the large value of the collision number means that even gross errors in it make a negligible contribution to the thermal conductivity. The results of the calculations of the thermal conductivity of the mixtures of hydrogen and nitrogen are contained in table 1 1 together with the Hirschfelder-Eucken results (AA = 0) and the experimental data at zero density. First, note that the inelasticM. J. ASSAEL A N D W . A. WAKEHAM 193 TABLE IO.-DATA EMPLOYED FOR THE CALCULATION OF THE THERMAL CONDUCTIVITY OF HYDROGEN -!- NITROGEN MIXTURES AT T,,, = 35 OC A,,(mon)/mW m-l K-l 141.0 AN2(mon)/mW m-' K-' 18.93 AHlNZ(mon)/mW m-l K-l 73.70 cN2Hz 200 cN2Nz 5.7 200 5.7 4 H 2 1.102 B k H 2 1.091 TABLE 1 1 .-COMPARISON OF THE CALCULATED THERMAL CONDUCTIVITY WITH THE EXPERIMENTAL DATA FOR HYDROGEN+ NITROGEN MIXTURES AT T,,, = 35 O C ~~ hydrogen mole fraction, Hirschfelder-Eucken, full eqn (4), experimental value, Acalc/mW m-lK-l Acalc/mWm-lK-l A,,p,/mWm-lK-l 0.21 36 0.4865 0.7338 43.60 74.15 116.2 43.79 73.96 114.9 44.02 74.33 115.2 term of eqn (4) contributes as much as 1.3 % to the mixture thermal conductivity so that its inclusion in the calculation is essential.Secondly, the deviations of the calculated total thermal conductivity of the mixtures from the experimental data amount to as much as 0.7%. Although this deviation constitutes reasonable agreement, it exceeds the estimated experimental error and, furthermore, the calculation con- sistently underestimates the thermal conductivity. The latter observation is a general characteristic of first-order kinetic theory approximation^^^ and supports the earlier finding that eqn (4) is not as accurate as the best experimental data.8 It would be possible to secure improved agreement with the present experimental data by means of modest, arbitrary adjustment of some of the parameters listed in table 10.For example, numerical experimentation reveals that the calculated thermal conductivity is particularly sensitive to cN2H2, so that it would be possible to determine this quantity from the experimental data. However, the inaccuracy of eqn (4) implies that such an evaluation would be burdened with a systematic error. For this reason we postpone the calculation of the collision number until a more accurate theoretical expression for the thermal conductivity becomes available.ELEVATED DENSITIES Mason and his collaborator~~~ have devised a scheme for the calculation of the thermal conductivity of gas mixtures based upon the modified Thorne-Enskog theory. Their equations reduce to the Hirschfelder-Eucken equation in the limit ofzero-density. Consequently we prefer here to use their procedure to evaluate the coefficient c1 in the density expansion of the thermal conductivity [eqn (3)], because this avoids the systematic error in the absolute thermal conductivity which is incurred as a result of the neglect of the inelastic contributions. Because the calculation again requires the use of dilute gas viscosity data, we are once more limited to a study of the hydrogen + nitrogen system.In order to carry out the calculation in the manner described by Kestin and Wakeham,25 we have made use of the thermal conductivity of the pure gases194 THERMAL CONDUCTIVITY OF BINARY GAS MIXTURES 1000 10 0 0.25 0.50 0.75 1.0 hydrogen mole fraction, XH* FIG. 3.-First density coefficient of thermal conductivity for mixtures of hydrogen +nitrogen. 0, Experimental results; -, calculated using the theory of Mason et aLZ4 The error bars correspond to 2.5 times the standard deviation of the measured coefficient. determined earlier,6 together with the viscosity data of Kestin and Yata.13 In addition, the molecular size parameters for the three interactions in the gas have been obtained from the second virial coefficient tabulations of Dymond and Smith." The calculated first density coefficients are compared with the experimental values in fig.3. The agreement displayed is satisfactory, particularly with regard to the weak dependence of the coefficient on composition in nitrogen-rich mixtures, and confirms the usefulness of the prediction scheme. The work described in this paper has been carried out with financial support from the S.R.C. M. J. Assael, M. Dix, A. Lucas and W. A. Wakeham, J. Chem. Soc., Faraday Trans. 1 , 1981,77,439. J. Kestin, R. Paul, A. A. Clifford and W. A. Wakeham, Physica (The Hague), 1980, 100A, 349. A. A. Clifford, R. Fleeter, J. Kestin and W. A. Wakeham, Physica (The Hague), 1979, %A, 467. A. A. Clifford, J. Kestin and W. A. Wakeham, Physica (The Hague), 1979, 97A, 298.R. Fleeter, J. Kestin and W. A. Wakeham, Physica (The Hague), 1980, 103A, 521. M. J. Assael and W. A. Wakeham, J. Chem. SOC., Faraday Trans. I , 1981, 77, 697. M. J. Assael and W. A. Wakeham, Ber. Bunsenges. Phys. Chem., 1980, 84, 840. J. J. Healy, J. J. de Groot and J. Kestin, Physica (The Hague), 1976, 82C, 392. ' R. Fleeter, J. 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